[Election-Methods] Partisan Politics

[Abd ul-Rahman Lomax]

At 05:33 PM 5/8/2008, Juho wrote:
(If there are e.g. two parties, one small and one large, the
>probability of getting two small party supporters (that would elect
>one of them to the next higher level) in a group of three is so small
>that in the next higher level the number of small party supporters is
>probably lower than at this level.)

Okay, let's do the math. Suppose the ratio of voters who are of some 
group is p, where 0 < p < 1. If x is not-p, then the permutations and 
probabilities for the four possibilities of 0 members, 1 member, 2 
members, and three members, are:

xxx, (1-p)^3 = P(0)
xxp, xpx, pxx, 3 * (1 - p)^2 * p = P(1)
xpp, pxp, ppx, 3 * (1 - p)* p^2 = P(2)
ppp, p^3 = P(3)

expanding those,
P(0) = 1 - 3p +3p^2 -p^3
P(1) = 3p -6p^2 +3p^3
P(2) = 3p^2 - 3p^3
P(3) = p^3.

To check, the sum simplifies to 1. These four are the only possibilities.

If the group selects based on majority p, then we have a p choice 
with P(2) and P(3). That occurs with probability

3p^2 -2p^3.

If p = 0.1, then the probability of a group choosing a p 
representative is .03 - .002 equals .028.

p is 10% of the population, but is represented in the next layer with 
only 2.8% of the elected representatives. And then the same 
phenomenon occurs in the next layer, etc., with the proportion of p 
declining more rapidly with each layer. I get 0.23% for the next 
layer. With many layers, as is necessary for this system to represent 
a large population the proportion of p rapidly approaches zero, and 
it becomes extraordinarily unlikely for the minority to be 
represented at all, even with an Assembly of, say, 100 members or 
more. And that is already a fairly large assembly, in my opinion. 
Assemblies that large tend to function mostly in committee.


Now, perhaps my math is wrong, I'm rusty and all that, and I make 
mistakes even when I understand clearly what to do. Mr. Gohlke, do 
you care to look at this?